Journal of Statistical Physics

, Volume 155, Issue 2, pp 323–391 | Cite as

A Brownian Particle in a Microscopic Periodic Potential



We study a model for a massive test particle in a microscopic periodic potential and interacting with a reservoir of light particles. In the regime considered, the fluctuations in the test particle’s momentum resulting from collisions typically outweigh the shifts in momentum generated by the periodic force, so the force is effectively a perturbative contribution. The mathematical starting point is an idealized reduced dynamics for the test particle given by a linear Boltzmann equation. In the limit that the mass ratio of a single reservoir particle to the test particle tends to zero, we show that there is convergence to the Ornstein–Uhlenbeck process under the standard normalizations for the test particle variables. Our analysis is primarily directed towards bounding the perturbative effect of the periodic potential on the particle’s momentum.


Brownian limit Linear Boltzmann equation Ornstein-Uhlenbeck process Nummelin splitting 



We are grateful to Professor Höpfner for sending a copy of Touati’s unpublished paper and giving helpful comments. We also thank an anonymous referee for offering many useful suggestions towards improving the presentation of this article. This work is supported by the European Research Council Grant No. 227772 and NSF Grant DMS-08446325.


  1. 1.
    Adams, C.S., Sigel, M., Mlynek, J.: Atom optics. Phys. Rep. 240, 143–210 (1994)ADSCrossRefGoogle Scholar
  2. 2.
    Athreya, K.B., Ney, P.: A new approach to the limit theory of recurrent Markov chains. Trans. Am. Math. Soc. 245, 493–501 (1978)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Brunnschweiler, A.: A connection between the Boltzmann equation and the Ornstein–Uhlenbeck process. Arch. Ration. Mech. Anal. 76, 247–263 (1981)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Birkl, G., Gatzke, M., Deutsch, I.H., Rolston, S.L., Phillips, W.D.: Bragg scattering from atoms in optical lattices. Phys. Rev. Lett. 75, 2823–2827 (1998)ADSCrossRefGoogle Scholar
  5. 5.
    Chung, K.L.: A Course in Probability Theory. Academic Pres, New York (1976)Google Scholar
  6. 6.
    Clark, J.T.: Suppressed dispersion for a randomly kicked quantum particle in a Dirac comb. J. Stat. Phys. 150, 940–1015 (2013)ADSCrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Clark, J.T.: A limit theorem to a time-fractional diffusion. Lat. Am. J. Probab. Math. Stat. 10(1), 117–156 (2013)MATHMathSciNetGoogle Scholar
  8. 8.
    Clark, J., Dubois, L.: Bounds for the state-modulated resolvent of a linear Boltzmann generator. J. Phys. A 45, 225207 (2012)ADSCrossRefMathSciNetGoogle Scholar
  9. 9.
    Clark, J., Maes, C.: Diffusive behavior for randomly kicked Newtonian particles in a periodic medium. Commun. Math. Phys. 301, 229–283 (2011)ADSCrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Dürr, D., Goldstein, S., Lebowitz, J.L.: A mechanical model for a Brownian motion. Commun. Math. Phys. 78, 507–530 (1981)ADSCrossRefMATHGoogle Scholar
  11. 11.
    Dzhaparidze, K., Valkeila, E.: On the Hellinger type distances for filtered experiments. Probab. Theory Relat. Fields 85, 105–117 (1990)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Freidlin, M.I., Wentzell, A.D.: Random perturbations of Hamiltonian systems. Mem. Am. Math. Soc. 109(523) (1994)Google Scholar
  13. 13.
    Friedman, N., Ozeri, R., Davidson, N.: Quantum reflection of atoms from a periodic dipole potential. J. Opt. Soc. Am. B 15, 1749–1755 (1998)ADSCrossRefGoogle Scholar
  14. 14.
    Hall, P., Heyde, C.C.: Martingale Limit Theory and its Application. Academic Press, New York (1980)MATHGoogle Scholar
  15. 15.
    Hennion, H.: Sur le mouvement d’une particule lourde soumise à des collisions dans un système infini de particules légères. Z. Wahrscheinlichkeitstheorie Verw. Geb. 25, 123–154 (1973)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Holley, R.: The motion of a heavy particle in a one dimensional gas of hard spheres. Probab. Theory Relat. Fields 17, 181–219 (1971)MathSciNetGoogle Scholar
  17. 17.
    Höpfner, R., Löcherbach, E.: Limit theorems for null recurrent Markov processes. Mem. Am. Math. Soc. 161 (2003)Google Scholar
  18. 18.
    Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer, Berlin (1987)CrossRefMATHGoogle Scholar
  19. 19.
    Kunze, S., Dürr, S., Rempe, G.: Bragg scattering of slow atoms from a standing light wave. Europhys. Lett. 34, 343–348 (1996)ADSCrossRefGoogle Scholar
  20. 20.
    Komorowski, T., Landim, C., Olla, S.: Fluctuations in Markov Processes. Springer, Berlin (2012)CrossRefMATHGoogle Scholar
  21. 21.
    Löcherbach, E., Loukianova, D.: On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions. Stoch. Processes Appl. 118, 1301–1321 (2008)CrossRefMATHGoogle Scholar
  22. 22.
    McClelland, J.J.: Atom-optical properties of a standing-wave light field. J. Opt. Soc. Am. B 12, 1761–1768 (1995)ADSCrossRefGoogle Scholar
  23. 23.
    Meyn, S.P., Tweedie, R.L.: Generalized resolvents and Harris recurrence of Markov processes. Contemp. Math. 149, 227–250 (1993)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Montroll, E.W., Weiss, G.H.: Random walks on lattices, II. J. Math. Phys. 6, 167–181 (1965)ADSCrossRefMathSciNetGoogle Scholar
  25. 25.
    Morsch, O.: Dynamics of Bose–Einstein condensates in optical lattices. Rev. Mod. Phys. 78, 179–215 (2006)ADSCrossRefGoogle Scholar
  26. 26.
    Nelson, E.: Dynamical Theories of Brownian Motion. Princeton University Press, Princeton (1967)MATHGoogle Scholar
  27. 27.
    Neveu, J.: Potentiel Markovien récurrent des chaînes de Harris. Ann. Inst. Fourier 22, 7–130 (1972)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Nummelin, E.: A splitting technique for Harris recurrent Markov chains. Z. Wahrsheinlichkeitstheorie Verw. Geb. 43, 309–318 (1978)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Pollard, D.: Convergence of Stochastic Processes. Springer, New York (1984)CrossRefMATHGoogle Scholar
  30. 30.
    Spohn, H.: Large Scale Dynamics of Interacting Particles. Springer, Berlin (1991)CrossRefMATHGoogle Scholar
  31. 31.
    Szász, D., Tóth, B.: Towards a unified dynamical theory of the Brownian particle in an ideal gas. Commun. Math. Phys. 111, 41–62 (1987)ADSCrossRefMATHGoogle Scholar
  32. 32.
    Touati, A.: Théorèmes limites pour les processus de Markov récurrents. C. R. Acad. Sci. Paris Sér. I Math. 305(19), 841–844 (1987)MATHMathSciNetGoogle Scholar
  33. 33.
    Uhlenbeck, G.E., Ornstein, L.S.: On the theory of Brownian motion. Phys. Rev. 36, 823–841 (1930)ADSCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA
  2. 2.Department of MathematicsUniversity of HelsinkiHelsinkiFinland

Personalised recommendations